Our final objective is to introduce scalable tools for learning a valid image density with an EBM. The work [1]} revealed that, in the absence of careful implementation, EBM learning always results in an unexpected outcome where steady-state samples from the learned density \(p(x;\theta )\)  have an oversaturated and unrealistic appearance that differs drastically from shortrun samples used during learning. This outcome affects all EBMs that are not specifically trained to overcome this defect, as well as related models such as normalizing flows, score-based models, and diffusion models (see Appendix REF ). To our knowledge, across different generative models, the only way to learn a well-formed potential energy surface approximating a complex density \(q(x)\)  that is compatible with efficient MCMC sampling is using an EBM.
